Mathematics – Representation Theory
Scientific paper
2008-11-18
Mathematics
Representation Theory
63 pages. v2: Title changed since the first part of this article can now be found as an independent article under the initial
Scientific paper
Let $Q$ be a finite quiver without oriented cycles and $\mathcal A(Q)$ be the coefficient-free cluster algebra with initial seed $(Q,\textbf u)$. Using the Caldero-Chapoton map, we introduce and investigate a family of generic variables in $\Z[\textbf u^{\pm 1}]$ containing the cluster monomials of $\mathcal A(Q)$. The aim of these generic variables is to give an explicit new method for constructing $\Z$-bases in the cluster algebra $\mathcal A(Q)$. If $Q$ is an affine quiver with minimal imaginary root $\delta$, we investigate differences between cluster characters associated to indecomposable representations of dimension vector $\delta$. We define the notion of \emph{difference property} which gives an explicit description of these differences. We prove in particular that this property holds for quivers of affine type $\tilde A$. When $Q$ satisfies the difference property, we prove that generic variables span the cluster algebra $\mathcal A(Q)$. If $\mathcal A(Q)$ satisfies some gradability condition, we prove that generic variables are linearly independent over $\mathbb Z$ in $\mathcal A(Q)$. In particular, this implies that generic variables form a $\Z$-basis in a cluster algebra associated to an affine quiver of type $\tilde A$.
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